Problem 90 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ A=P\left(1+\frac{i}{100}\right)^{2} ; \quad \text { for } i $$

Step-by-Step Solution

Answer

\( i = 100\left(\sqrt{\frac{A}{P}} - 1\right) \)

Step 1: Objective Clarification

Our task is to solve for the variable \( i \) in the equation \( A = P\left(1+\frac{i}{100}\right)^{2} \). This means isolating \( i \) on one side of the equation.

Step 2: Isolate the Square Term

Start by dividing both sides of the equation by \( P \) to remove it from the right side: \[ \frac{A}{P} = \left(1+\frac{i}{100}\right)^{2} \]. Now, we have \( (1+\frac{i}{100})^{2} \) isolated.

Step 3: Eliminate the Square

Take the square root of both sides to eliminate the square. This gives: \[ \sqrt{\frac{A}{P}} = 1 + \frac{i}{100} \]. Ensure considering both the positive and negative roots.

Step 4: Rearrange to Solve for i

Subtract 1 from both sides to separate \( \frac{i}{100} \): \[ \frac{i}{100} = \sqrt{\frac{A}{P}} - 1 \].

Step 5: Finalize the Solution for i

Multiply both sides by 100 to solve for \( i \): \[ i = 100\left(\sqrt{\frac{A}{P}} - 1\right) \]. This is your expression for \( i \).

Key Concepts

Isolation of Variables

In solving equations, one of the most important skills is isolating the variable in question. The goal is to have the variable alone on one side of the equation, which stands for finding the simplest form where it equals an expression involving only known quantities. This typically involves performing arithmetic operations that strategically remove other terms and coefficients from around the variable.

• Identify what you need to solve for. In our exercise, this meant recognizing that we were solving for \( i \).
• Move any terms that do not include the variable to the opposite side of the equation via addition or subtraction.
• Address any coefficients or factors by using division or multiplication to 'undo' these operations.

By dividing both sides of the equation by \( P \), we successfully isolated the quantity \((1+\frac{i}{100})^2\) from other elements in the equation. This careful step-by-step manipulation is crucial for an accurate solution.

Taking Square Roots

Once you've isolated a term like \((1+\frac{i}{100})^2\), the next move is typically to take the square root, especially if the term is a squared expression. Removing the square is necessary to simplify the equation further.Taking the square root can be tricky because it introduces two possible values: the positive and negative roots. Keep this duality in mind because it can impact your final solution significantly, and it is fundamental to ensure all possibilities are covered.

• Clearly state that you will take the square root on both sides of the equation to maintain balance.
• Write down both potential answers when applicable, showing that both the positive and negative roots have been considered.

This practice of considering both roots is essential in avoiding any casual mistakes and provides a deeper understanding of how square manipulations influence an equation.

Algebraic Manipulation

Algebraic manipulation is about understanding the properties of numbers and operations to rearrange and simplify equations. After isolating the squared term and taking its square root in our equation, we're left with a more straightforward linear expression.Algebraic manipulation in this context involves isolating the desired variable, \( i \), from within the expression \( 1+\frac{i}{100} \). You achieve this by performing simple operations such as addition, subtraction, multiplication, and division:

• Subtract a number to simplify and isolate the specific variable part.
• Multiply or divide to eliminate fractions or coefficients.

For instance, after isolating \( \sqrt{\frac{A}{P}} = 1 + \frac{i}{100} \), subtracting 1 gives us \( \frac{i}{100} = \sqrt{\frac{A}{P}} - 1 \). Multiplying the result by 100 finalizes the expression for \( i \). Mastering these manipulations can be a powerful tool in cracking any complex algebraic problem.